Distance lecture

Distance and Dimension Reduction

Introduction

The concept of distance is quite intuitive. For example, when we cluster animals into subgroups, we are implicitly defining a distance that permits us to say what animals are “close” to each other.

Many of the analyses we perform with high-dimensional data relate directly or indirectly to distance. Many clustering and machine learning techniques rely on being able to define distance, using features or predictors. For example, to create heatmaps, which are widely used in genomics and other highthroughput fields, a distance is computed explicitly.

In these plots the measurements, which are stored in a matrix, are
represented with colors after the columns and rows have been
clustered. (A side note: red and green, a common color theme for heatmaps,
are two of the most difficult colors for many color-blind people to discern.)
Here we will learn the necessary mathematics and computing
skills to understand and create heatmaps. We start by reviewing the
mathematical definition of distance.

Euclidean Distance

As a review, let’s define the distance between two points, and , on a Cartesian plane.

The euclidean distance between and is simply:

Distance in High Dimensions

We introduce a dataset with gene expression measurements for 22,215 genes from 189 samples. The R objects can be downloaded like this:

We are interested in describing distance between samples in the context of this dataset. We might also be interested in finding genes that behave similarly across samples.

To define distance, we need to know what the points are since mathematical distance is computed between points. With high dimensional data, points are no longer on the Cartesian plane. Instead they are in higher dimensions. For example, sample is defined by a point in 22,215 dimensional space: . Feature is defined by a point in 189 dimensions

Once we define points, the Euclidean distance is defined in a very similar way as it is defined for two dimensions. For instance, the distance between two samples and is:

and the distance between two features and is:

Distance with matrix algebra

The distance between samples and can be written as

with and columns and . This result can be very convenient in practice as computations can be made much faster using matrix multiplication.

Examples

We can now use the formulas above to compute distance. Let’s compute distance between samples 1 and 2, both kidneys, and then to sample 87, a colon.

x<-e[,1]y<-e[,2]z<-e[,87]sqrt(sum((x-y)^2))

## [1] 85.8546

sqrt(sum((x-z)^2))

## [1] 122.8919

As expected, the kidneys are closer to each other. A faster way to compute this is using matrix algebra:

sqrt(crossprod(x-y))

## [,1]
## [1,] 85.8546

sqrt(crossprod(x-z))

## [,1]
## [1,] 122.8919

Now to compute all the distances at once, we have the function dist. Because it computes the distance between each row, and here we are interested in the distance between samples, we transpose the matrix

d<-dist(t(e))class(d)

## [1] "dist"

Note that this produces an object of class dist and, to access the entries using row and column, indexes we need to coerce it into a matrix:

as.matrix(d)[1,2]

## [1] 85.8546

as.matrix(d)[1,87]

## [1] 122.8919

It is important to remember that if we run dist on e, it will compute all pairwise distances between genes. This will try to create a matrix that may crash your R sessions.